ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  seqeq1 Unicode version

Theorem seqeq1 10214
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
Assertion
Ref Expression
seqeq1  |-  ( M  =  N  ->  seq M (  .+  ,  F )  =  seq N (  .+  ,  F ) )

Proof of Theorem seqeq1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . . 6  |-  ( M  =  N  ->  M  =  N )
2 fveq2 5414 . . . . . 6  |-  ( M  =  N  ->  ( F `  M )  =  ( F `  N ) )
31, 2opeq12d 3708 . . . . 5  |-  ( M  =  N  ->  <. M , 
( F `  M
) >.  =  <. N , 
( F `  N
) >. )
4 freceq2 6283 . . . . 5  |-  ( <. M ,  ( F `  M ) >.  =  <. N ,  ( F `  N ) >.  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )  = frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. N , 
( F `  N
) >. ) )
53, 4syl 14 . . . 4  |-  ( M  =  N  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )  = frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. N , 
( F `  N
) >. ) )
6 fveq2 5414 . . . . . 6  |-  ( M  =  N  ->  ( ZZ>=
`  M )  =  ( ZZ>= `  N )
)
7 eqid 2137 . . . . . 6  |-  _V  =  _V
8 mpoeq12 5824 . . . . . 6  |-  ( ( ( ZZ>= `  M )  =  ( ZZ>= `  N
)  /\  _V  =  _V )  ->  ( x  e.  ( ZZ>= `  M
) ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. )  =  (
x  e.  ( ZZ>= `  N ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) )
96, 7, 8sylancl 409 . . . . 5  |-  ( M  =  N  ->  (
x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  ( ZZ>= `  N ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) )
10 freceq1 6282 . . . . 5  |-  ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  ( ZZ>= `  N ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. N ,  ( F `  N ) >. )  = frec ( ( x  e.  ( ZZ>= `  N ) ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. N , 
( F `  N
) >. ) )
119, 10syl 14 . . . 4  |-  ( M  =  N  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. N ,  ( F `  N ) >. )  = frec ( ( x  e.  ( ZZ>= `  N ) ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. N , 
( F `  N
) >. ) )
125, 11eqtrd 2170 . . 3  |-  ( M  =  N  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )  = frec ( ( x  e.  ( ZZ>= `  N ) ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. N , 
( F `  N
) >. ) )
1312rneqd 4763 . 2  |-  ( M  =  N  ->  ran frec ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  =  ran frec ( ( x  e.  (
ZZ>= `  N ) ,  y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. N , 
( F `  N
) >. ) )
14 df-seqfrec 10212 . 2  |-  seq M
(  .+  ,  F
)  =  ran frec (
( x  e.  (
ZZ>= `  M ) ,  y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
15 df-seqfrec 10212 . 2  |-  seq N
(  .+  ,  F
)  =  ran frec (
( x  e.  (
ZZ>= `  N ) ,  y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. N , 
( F `  N
) >. )
1613, 14, 153eqtr4g 2195 1  |-  ( M  =  N  ->  seq M (  .+  ,  F )  =  seq N (  .+  ,  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   _Vcvv 2681   <.cop 3525   ran crn 4535   ` cfv 5118  (class class class)co 5767    e. cmpo 5769  freccfrec 6280   1c1 7614    + caddc 7616   ZZ>=cuz 9319    seqcseq 10211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-iota 5083  df-fv 5126  df-oprab 5771  df-mpo 5772  df-recs 6195  df-frec 6281  df-seqfrec 10212
This theorem is referenced by:  seqeq1d  10217  seq3f1olemqsum  10266  seq3id  10274  seq3z  10277  iserex  11101  summodclem2  11144  summodc  11145  zsumdc  11146  isumsplit  11253  ntrivcvgap  11310  ntrivcvgap0  11311  ege2le3  11366
  Copyright terms: Public domain W3C validator